hello

Unlike what you would think the two words are a lot different in the meaning

Gateau comes from the German wastel and it means cake (to summarize)

Gater comes from old French and it's actually a pejorative word(that's why I said the 2 words are different)

Gater means to spoil and more precisely to ruin

gater un enfant = to spoil a kid = actually means to ruin him

we also say : la grêle a gater la vigne =hail ruined the vine

Even in english you might hear : don't eat before dinner you will spoil your appetite

meaning you will ruin your appetite

I hope this helps you

maybe there is no connection between them but you can create a connection to memorize vocabulary for example you say : cet enfant est trop gâté, son père lui achète trop de gâteaux

"gater" in French has same meaning as "to spoil" in English:

"Un gateau" is a sweet cake, sweet treat to treat people and yourself in a nice way.

That is why i think that 'gater" (to overtreat kids) has connection with "un gateau" (a sweet treat). It is possible Gater les enfants (ou les gens) WITH les gateau...

Un gâteau peut se gâter si personne ne le mange à temps.

-> A cake can spoil if nobody eats it on time.